A342531 - OEIS

A342531

Triangle read by rows where T(n,k) is the number of strict integer partitions of n with maximal descent k, n >= 0, 0 <= k <= n.

1, 1, 0, 1, 0, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 1, 0, 2, 1, 1, 0, 1, 0, 0, 1, 2, 1, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 2, 1, 1, 1, 0, 1, 0, 0, 1, 1, 2, 3, 1, 1, 1, 1, 0, 1, 0, 0

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OFFSET

0,39

COMMENTS

The maximal descent of an empty or singleton partition is considered to be 0.

LINKS

Table of n, a(n) for n=0..77.

Gus Wiseman, Sequences counting and ranking integer partitions by the differences of their successive parts.

EXAMPLE

Triangle begins:

1 0

1 0 0

1 1 0 0

1 0 1 0 0

1 1 0 1 0 0

1 1 1 0 1 0 0

1 1 1 1 0 1 0 0

1 0 2 1 1 0 1 0 0

1 2 1 1 1 1 0 1 0 0

1 1 2 2 1 1 1 0 1 0 0

1 1 2 3 1 1 1 1 0 1 0 0

1 1 3 2 3 1 1 1 1 0 1 0 0

1 1 3 3 3 2 1 1 1 1 0 1 0 0

1 1 3 4 3 3 2 1 1 1 1 0 1 0 0

1 3 3 4 4 3 2 2 1 1 1 1 0 1 0 0

1 0 5 5 5 4 3 2 2 1 1 1 1 0 1 0 0

1 1 4 7 5 5 4 2 2 2 1 1 1 1 0 1 0 0

1 2 5 6 7 6 4 4 2 2 2 1 1 1 1 0 1 0 0

1 1 5 9 7 7 6 4 3 2 2 2 1 1 1 1 0 1 0 0

1 1 6 9 9 7 8 5 4 3 2 2 2 1 1 1 1 0 1 0 0

Row n = 15 counts the following strict partitions (empty columns indicated by dots, A..F = 10..15):

F 87 753 96 762 A5 A41 B4 B31 C3 C21 D2 . E1 . .

654 6432 852 843 861 9321 A32

54321 6531 7431 951 942

7521 8421

MATHEMATICA

Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&If[Length[#]<=1, k==0, Max[Differences[Reverse[#]]]==k]&]], {n, 0, 15}, {k, 0, n}]

CROSSREFS

The non-strict version is A238353.

A000041 counts partitions (strict: A000009).

A049980 counts strict partitions with equal differences.

A325325 counts partitions with distinct differences (ranking: A325368).

A325545 counts compositions with distinct differences.

Cf. A003114, A003242, A005117, A034296, A049988, A238710, A342098, A342514.

Sequence in context: A145679 A007273 A016319 * A287156 A092510 A117208

Adjacent sequences: A342528 A342529 A342530 * A342532 A342533 A342534

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Mar 25 2021

STATUS

approved